Finite Group Databasebeta

← #13052663

Subgroups — #13052663

Order 1344.

224 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#6450C211yesmaximal, characteristic, complemented
2#10620935C211yesmaximal, characteristic
3C2×#66570C211yesmaximal, characteristic, complemented
4C2×#66564C211yesmaximal, characteristic
5#10620649C211yesmaximal, characteristic, complemented
6C7×#809C211yesmaximal, characteristic
7Dic7⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89539—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C3×Dic7⋊C4—21no—
16C3×C4×Dic7—21no—
17C3×C4×Dic7C2211yescharacteristic
18(C2×D3)⋊Dic7—21no—
19(C2×D3)⋊Dic7—21no—
20#66570—22no—
21#66564—22no—
22#6499—31no—
23C2×#1142—31no—
24C2×Dic28—31no—
25#6450D311yescharacteristic, complemented
26C7×#102—31no—
27Dic7⋊C8—31no—
28#6384—31no—
29#5028—71nomaximal, Hall
30C22×D3×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34Dic42—21no—
35Dic42—21no—
36C7×D12—21no—
37C7×D12—21no—
38C2×C3×Dic7—21no—
39C2×C3×Dic7D411yescharacteristic
40C2×C3×Dic7D411yescharacteristic
41C3×Dic14—21no—
42C3×Dic14D412yes—
43C3⋊(C7⋊C8)—21no—
44C7×C3⋊C8—21no—
45C2×C7×D4—31no—
46C2×Dic14—31no—
47C2×Dic14C2×D311yescharacteristic
48C2×C7×C8—31no—
49C7×C4⋊C4C2×D311yescharacteristic
50C22⋊Dic7—61no—
51C22⋊Dic7—61no—
52#1142—62no—
53Dic7⋊C4—21no—
54C4×Dic7—21no—
55C4×Dic7C2×D311yescharacteristic
56C2×C7⋊C8—31no—
57Dic28—62no—
58C3×C4×Q8—71no—
59C2×#216—71no—
60C2×#215—71no—
61#875—71no—
62#810—71no—
63#809D711yescharacteristic
64C4⋊(C3⋊C8)—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D3×C7—41no—
67C2×D3×C7—41no—
68C2×D3×C7—41no—
69C3×C4×C7—41no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—41no—
73C3×Dic7—22no—
74C3×Dic7—21no—
75C3×Dic7—21no—
76#464—211noSylow
77C23×C7—61no—
78C7×D4—61no—
79C7×D4—61no—
80C2×C4×C7—21no—
81C2×C4×C7C22×D311yescharacteristic
82C2×Dic7—21no—
83C2×Dic7—61no—
84C2×Dic7#6711yescharacteristic
85C2×Dic7#6711yescharacteristic
86Dic14—21no—
87Dic14—61no—
88Dic14—61no—
89Dic14D1212yes—
90C7×C8—61no—
91C7⋊C8—61no—
92C2×C3×Q8—71no—
93C2×D12C2×D711yescharacteristic
94C2×Dic6—71no—
95C3×C4⋊C4—141no—
96C3×C4⋊C4C2×D711yescharacteristic
97C3×C42—71no—
98C3×C42—141no—
99#216—142no—
100#215—142no—
101(C2×D3)⋊C4—141no—
102(C2×D3)⋊C4—141no—
103C2×C3⋊C8—71no—
104C2×C3⋊C8C2×D711yescharacteristic
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7#13511yescharacteristic
107C2×C3×C7#13711yescharacteristic
108D3×C7—81no—
109C2×Q16—211no—
110C2×SD16—211no—
111#124—211no—
112C4×Q8—71no—
113C4⋊C8—211no—
114#103—211no—
115#102—31no—
116C22×C7—121no—
117C22×C7—121no—
118C22×C7—121no—
119C22×C7C2×#6711yescharacteristic
120C4×C7—41no—
121C4×C7C2×D1211yescharacteristic
122C4×C7#23511yescharacteristic
123Dic7—22no—
124Dic7—21no—
125Dic7—21no—
126Dic7—121no—
127C22×D3—21no—
128C3×Q8—72no—
129C3×Q8—141no—
130C2×C3×C4—21no—
131C2×C3×C4—71no—
132C2×C3×C4—71no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—141no—
136D12—21no—
137D12—21no—
138Dic6—141no—
139Dic6—141no—
140C3⋊C8—21no—
141C3⋊C8—141no—
142C3×C7#46411yescharacteristic, complemented, Hall
143C2×Q8—71no—
144C2×Q8—211no—
145C2×D4—31no—
146Q16—422no—
147SD16—422no—
148C2×C8—31no—
149C2×C8—211no—
150C4⋊C4—141no—
151C4⋊C4D3×D711yescharacteristic
152C22⋊C4—421no—
153C22⋊C4—421no—
154C42—71no—
155C42—141no—
156C2×C7—241no—
157C2×C7#93011yescharacteristic
158C2×C7#94211yescharacteristic
159C2×C7#95011yescharacteristic
160C22×C3D4×D711yescharacteristic
161C2×D3—41no—
162C2×D3—41no—
163C2×D3—41no—
164C3×C4—41no—
165C3×C4—142no—
166C3×C4—141no—
167C3×C4—141no—
168C3×C4#115711yescharacteristic
169C3×C4D4×D711yescharacteristic
170Dic3—281no—
171C23—61no—
172Q8—72no—
173Q8—141no—
174Q8—421no—
175Q8—421no—
176D4—61no—
177D4—61no—
178C2×C4—21no—
179C2×C4—71no—
180C2×C4—71no—
181C2×C4—141no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—421no—
186C7#502811yescharacteristic, complemented, Sylow
187C2×C3#644711yescharacteristic
188C2×C3#647511yescharacteristic
189C2×C3#647811yescharacteristic
190D3—81no—
191C22—121no—
192C22—121no—
193C22—121no—
194C22D7×#6711yescharacteristic
195C4—41no—
196C4—142no—
197C4—141no—
198C4—141no—
199C4—841no—
200C4#6667511yescharacteristic
201C4D7×D1211yescharacteristic
202C3#8953911yescharacteristic, complemented, Sylow
203C2—241no—
204C2#1062101011yescharacteristic
205C2#1062105211yescharacteristic
206C2#1062107511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×D12C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#464
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#6450C2semidirect
3. C2×#66570C2semidirect
5. #10620649C2semidirect
7. Dic7⋊(C3⋊C8)C2semidirect
25. #6450D3semidirect
142. C3×C7#464semidirect
186. C7#5028semidirect
202. C3#89539semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#64502
#106209352
C2×#665702
C2×#665642
#106206492
C7×#8092
Dic7⋊(C3⋊C8)2
#895393
#50287